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1
=
3
r
w
u
z
L
¼
1
:
31
3
ð
3
:
18
Þ
This means that the standard deviations of the horizontal wind components
Eq. (
3.17
) are height independent in the unstable surface layer while the standard
deviation of the vertical wind component (
3.18
) increases with height. Originally,
Panofsky et al. (
1977
) and Arya (
1995
) have given 12 as a common value for the
numbers 15.625 and 6.859 in (
3.17
). The different choice has been made here in
order to be consistent with the relations in (
3.9
) in the limit of neutral stratification.
Arya (
1995
) gives for the standard deviations of the 10 Hz fluctuations of the
wind components in the unstably stratified Ekman layer above the Prandtl layer:
r
u
;
v
;
w
¼
0
:
6w
ð
3
:
19
Þ
with the convective velocity scale:
w
¼
gz
i
H
w
0
H
0
ð
3
:
20
Þ
This convective velocity scale substitutes the friction velocity as a scaling
velocity in situations where vertical velocities due to unstable thermal stratification
are in the same order as the horizontal wind speeds. This means that the standard
deviation of the vertical velocity component increases with height in the unstable
Prandtl layer due to relation (
3.18
) and then stays constant above it due to relation
(
3.19
).
3.1.1.3 Stable Stratification
The stable type of the surface layer, which is characterized by a downward surface
heat flux (L
*
[ 0) and a stable thermal stratification of the air, is usually found at
night time, over waters that are colder than the air above, and over ice and snow-
covered surfaces. For positive values of z/L
*
, the correction functions for the
logarithmic wind profile read (Businger et al.
1971
; Dyer
1974
; Holtslag and de
Bruin
1988
):
8
<
:
az
=
L
for 0\z
=
L
0
:
5
W
m
ð
z
=
L
Þ¼
Az
=
L
þ
B
ð
z
=
L
C
=
D
Þ
exp
ð
Dz
=
L
Þþ
BC
=
D
ð
3
:
21
Þ
for 0
:
5
z
=
L
7
where a = 5, A = 1, B = 2/3, C = 5, and D = 0.35. The vertical wind profile
u(z) in the stable surface layer is then described again by Eq. (
3.16
) but now using
the functions (
3.21
) for stable stratification.
The air temperature in the stable boundary layer vertically decreases less than
the adiabatic lapse rate and the potential temperature (
3.8
) increases with height.
The standard deviations of the wind components are usually assumed to be constant
with height in the same way as described by (
3.9
) for the neutral ABL (Arya
1995
).
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